A system's input-output relationship can be classified as either linear or nonlinear. The input-output relationships in systems can typically be represented mathematically using equations referred to as transfer functions. Linear systems such as linear filters used in signal processing applications have linear transfer functions, therefore are typically easier to model than nonlinear systems.
A linear filter with an infinite impulse response can be generally described using the following equation:yn=h0un+. . . +hN−1un−N+1−g1yn−1−. . . −gN−1yn−N+1  (Equation 1),where un−j is the filter input at time n−j, yn−j is the filter output at time n−j, hk and gk are filter coefficients. A linear filter with a finite impulse response can be generally described asyn=h0un+. . . +hN−1un−N+1  (Equation 2).
Linear filters have states, which can be fully described using linear equations. One example of a state-space description is the following:Xn+1=AXn+Bun  (Equation 3),yn=CXn+Dun  (Equation 4),where Xn+l is the next-state vector at time n+1, Xn is the current state vector at time n, A is an N×N state-transition matrix, B is an N×1 vector that maps the input variable to the next-state, C is a 1×N vector that maps the current state to the output, D is a scalar that maps the input to the output.
The state-space description can be used to observe state variables that characterize the system. State variables are useful for controlling system properties such as noise and stability, and for providing better observability of system parameters. State variable description of linear systems is well developed and various implementations of the state-space models of linear systems exist. In contrast, state variables in nonlinear systems such as communication and control systems are often not directly accessible to the user. Furthermore, because of their deviation from the standard linear filters and linear systems, nonlinear systems typically do not offer an easy way to derive their state-space representations. As a result, it is often difficult to control nonlinear systems using state variables. Some of the existing techniques use averages to approximate the nonlinear relationships in order to derive state-space representations. However, the approximations are often inaccurate and complex to implement.
It would be desirable to have a better way to model the state variables in nonlinear systems, so that the systems' dynamics and stability could be better controlled. It would also be useful if the state variables of inverse nonlinear systems could be determined as well.